*Article* **Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method**

### **Tien Dat Pham 1, Quoc Hoa Pham 1, Van Duc Phan 2, Hoang Nam Nguyen 3,\* and Van Thom Do 1**


Received: 7 March 2019; Accepted: 7 May 2019; Published: 17 May 2019

**Abstract:** An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique for triangular shell element (MITC3), called ES-MITC3, for free vibration analysis of functionally graded shells is investigated in this work. In the formulation of the ES-MITC3, the sti ffness matrices are obtained by using the strain-smoothing technique over the smoothing domains that are formed by two adjacent MITC3 triangular shell elements sharing an edge. The strain-smoothing technique can improve significantly the accuracy and convergence of the original MITC3. The material properties of functionally graded shells are assumed to vary through the thickness direction by a power–rule distribution of volume fractions of the constituents. The numerical examples demonstrated that the present ES-MITC3method is free of shear locking and achieves the high accuracy compared to the reference solutions in the literature.

**Keywords:** FGMshells; edge-based smoothed finite element method (ES-FEM); mixed interpolation of tensorial components (MITC)
